Beyond Pythagoras
What this coursework has asked me to do is to investigate and find a
generalisation, for a family of Pythagorean triples. This will include
odd numbers and even numbers.
I am going to investigate a family of right-angled triangles for which
all the lengths are positive integers and the shortest is an odd
number.
I am going to check that the Pythagorean triples (5,12,13) and
(7,24,25) cases work; and then spot a connection between the middle
and longest sides.
The first case of a Pythagorean triple I will look at is:
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[IMAGE][IMAGE][IMAGE]The numbers 5, 12 and 13 satisfy the connection.
5² + 12² = 13²
25 + 144 = 169
169 = 13
The second case of a Pythagorean triple I will look at is:
[IMAGE][IMAGE]The numbers 7, 24 and 25 satisfy the connection.
7² + 24² = 25²
[IMAGE][IMAGE][IMAGE]49 + 576 = 625
625 = 25
There is a connection between the middle and longest side. This is
that there is a one number difference.
So if M= middle and L= longest
L = M + 1
I am going to use the triples, (3,4,5), (5,12,13) and (7,24,25) to
find other triples. Then I will put my results in a table and look for
a pattern that will occur. I will then try and predict the next
results in the table and prove it.
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n
Smallest
O
Middle
O
Longest
O
1
3
4
5
2
5
12
13
3
7
24
25
There is a clear pattern between the middle and longest side.
There is also a sequence forming.
n = 1 S = 3 M = 4 L = 5
n = 2 S = 5 M = 12 L = 13
8. Repeat from step 4 twice more so you end up with three results for
What is trigonometry? Well trigonometry, according to the Oxford Dictionary ‘the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.’ Here is a simplified definition of my own: Trigonometry is a division of mathematics involving the study of the relativity of angles and sides of triangles. The word trigonometry originated from the Latin word: trigonometria.
in 212 B.C. at the age of 75 in Syracuse. It is said that he was killed
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Drawing from Book VI of the Republic (pp. 154-160), Plato used the Allegory of the Cave (Book VII, pp. 160-161) as an ultimate example to illustrate the importance of the good discussed before and to eloquently conclude his line of thought. Although he never exactly said what the Good was, his representation of its functions and existence adequately compensated for this minor shortcoming. The following essay aims to analyze the passage by synthesizing its main ideas and incorporating them into the broader framework of Plato’s philosophy.
Archimedes, a name commonly associated with the beginning of science, was an engineer and one of the greatest mathematicians in history. His impact on modern science rests on his use of experiment and invention to test ideas and his use mathematics to describe the basic principles of physical phenomena.
Empedocles was born in Acragas, Sicily about 492 BCE to a distinguished and aristocratic family. His father, Meto, is believed to have been involved in overthrowing Thrasydaeus who was the tyrant of Agrigentum in the year 470 BCE. Empedocles is said to have been somewhat wealthy and was a popular politician and a champion of democracy and equality.
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are how to follow lines of reasoning, how to say precisely what is intended, and
I have conducted my research through interview with someone familiar with construction and development as how geometry is used in these fields.
yet we know relatively few facts of his life. We are not exactly sure of his birth
(XXY) – three unknowns, test is which is odd one out (Y = 1, Y = 2, or Y = 3).
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